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A351596
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Numbers k such that the k-th composition in standard order has all distinct run-lengths.
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5
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0, 1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 19, 21, 23, 26, 28, 30, 31, 32, 35, 36, 39, 42, 47, 56, 60, 62, 63, 64, 67, 71, 73, 74, 79, 84, 85, 87, 95, 100, 106, 112, 119, 120, 122, 123, 124, 126, 127, 128, 131, 135, 136, 138, 143, 146, 159, 164, 168, 170, 171
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OFFSET
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1,3
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COMMENTS
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The n-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
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LINKS
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EXAMPLE
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The terms together with their binary expansions and corresponding compositions begin:
0: 0 ()
1: 1 (1)
2: 10 (2)
3: 11 (1,1)
4: 100 (3)
7: 111 (1,1,1)
8: 1000 (4)
10: 1010 (2,2)
11: 1011 (2,1,1)
14: 1110 (1,1,2)
15: 1111 (1,1,1,1)
16: 10000 (5)
19: 10011 (3,1,1)
21: 10101 (2,2,1)
23: 10111 (2,1,1,1)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@Length/@Split[stc[#]]&]
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CROSSREFS
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The version using binary expansions is A044813.
The version for Heinz numbers and prime multiplicities is A130091.
The version for runs instead of run-lengths is A351290, counted by A351013.
A011782 counts integer compositions.
A085207 represents concatenation of standard compositions, reverse A085208.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct run-lengths:
- Number of distinct parts is A334028.
Cf. A098859, A106356, A175413, A238279, A242882, A328592, A329745, A333628, A350952, A351015, A351202.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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